Majorizing measures and proportional subsets of bounded orthonormal systems

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Abstract
In this article we prove that for any orthonormal system (\varphi_j)_{j=1}^n \subset L_2 that is bounded in L_{\infty} , and any 1 < k < n , there exists a subset I of cardinality greater than n-k such that on \mathrm{span}\{\varphi_i\}_{i \in I} , the L_1 norm and the L_2 norm are equivalent up to a factor \mu (\log \mu)^{5/2} , where \mu = \sqrt{n/k} \sqrt{\log k} . The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.

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